Abstract
We propose in this book a “tychastic viabilist” approach for solving such problems (1) taking into account the “viability” constraint that the value of portfolio is always above a floor (liabilities, variable annuities, etc. (2) using the “tychastic” approach to translate mathematically the concept of uncertainty.
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Notes
- 1.
Based on [33, 34, 37, 49, 50, 69, 149–151].
- 2.
Since European Options are nicknamed “vanilla options”, because their flavor is insipid and widely popular, we suggest to nickname this example as the “lychee
VPPI” example.
- 3.
The software of the VPPI Robot-Insurer of VIMADES has been registered on April 10, 2009, at the INPI, the French Institut National de la Propriété Industrielle.
- 4.
The European cousin of the American Fed Funds Effective (Overnight Rate) and the British London Inter-Bank Offered Rate (LIBOR), object of recent criminal manipulations.
- 5.
Even if it is not continuous, but “lower semicontinuous” (with jumps), which is the case of variable annuities.
- 6.
Or, in mathematical terms, that the evolution \(t \mapsto (t, W(t))\) is viable in the epigraph of the function \(L(\cdot )\). It is the subset \(\mathcal{E}p(L):= \{(t, W) \in \mathbb {R}^{2} \; \text{ such } \text{ that } \; W \ge L(t)\}\). Hence \((t, W(t))\) is viable in the epigraph of \(L\) if and only if, for all \(t \in [0, T]\), inequality \(W(t) \ge L(t)\) is satisfied.
- 7.
They are particular cases of the concept of quantitative tychastic risk measure of an environment under a tychastic system which is not ambiguous once the environment and the tychastic system are given.
- 8.
This allows the investor to revise at each date the lower bounds of the risky assets for computing the MGI by “rebalancing” the computation of the portfolio if he or her chooses to do so.
- 9.
How this wonderful word, which should exemplify, as in other Romance languages, impartiality, fairness, justice, rightfulness, not to mention the concept of ethics, came to mean the interest of shareholders in a company? When returns on equity around 15 % and more became standards after the years 1980, equities became really inequitable.
- 10.
This is the Bollinger percent index of the minimum guaranteed investment \(W^{\heartsuit }(t)\) in the cushion tube \([L(t), W(t)]\) (see Definition 2.1.2).
- 11.
It go back at least to 935 B.C., in the book of Ecclesiastes of the bible: “But divide your investments among many places, for you do not know what risks might lie ahead”. In China, the proverb
means that “A wily hare which has three burrows can keep itself safe”.
- 12.
The part of the public regulatory authority advocated by the Solvency II directive was abdicated in favor of private rating agencies.
- 13.
The exposure of an asset in the portfolio is the product of the number of shares of asset by the asset price. They could play the rôle of the “systematic risk” \(\beta \) measuring the sensitivity of the expected excess asset returns to the expected excess market returns in capital asset pricing types of models (CAPM) going back to the 1952 research of Harry Markowitz in [135].
- 14.
This “comb” \(t \mapsto C(t)\) the teeth (impulses) of which are the amounts \(C(t)\) at payment phases is only lower semicontinuous. The evolutionary engine is then impulsive, and the theory of impulse systems allows us to define their solutions. See Sect. 1.4 for another example of impulse systems and 12.3 [28].
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Aubin, JP., Chen, L., Dordan, O. (2014). The Viabilist Portfolio Performance and Insurance Approach. In: Tychastic Measure of Viability Risk. Springer, Cham. https://doi.org/10.1007/978-3-319-08129-8_1
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DOI: https://doi.org/10.1007/978-3-319-08129-8_1
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